Re: Sci-fi Alcubierre Drive and exotic matter/Casimir Effect

[Chris Hillman <hillman@math.washington.edu>]

On Wed, 5 Jun 2002, Becca Heisler wrote:

> I am a science fiction writer looking for a warp drive within the
> limits of general relativity.

You will be be disappointed, then!  No such thing is known, and currently
this circumstance appears unlikely change.  Indeed, even if gtr is wrong
(as in some sense it must be, because it is not a quantum theory--
however, many believe it almost surely must be the preferred classical
field theory limit of any viable quantum theory of gravity which may
happen along later in the new century), there are various arguments
suggesting that warp bubbles are physically impossible.  These arguments
are not completely compelling (at the level of say "perpetual motion is
impossible"), as far as I can see, but they are much more compelling than
the rather weak suggestions that warp bubbles -might- exist, which have
been long on speculative flights of the imagination, but very lacking in
specific physical mechanisms.

> I am VERY new to this field of physics and physics in general, so I
> have some questions: 1)Is the Alcubierre/Broeck Drive

First of all, it is essential to understand that the Alcubierre warp
bubble spacetime (1994)

  http://xxx.lanl.gov/abs/gr-qc/0009013

and the closely related Van Den Broeck warp bubble spacetime (1999)

  http://xxx.lanl.gov/abs/gr-qc/9905084

are Lorentzian manifolds, but they are -not- in any reasonable sense
"solutions" to the Einstein field equation (EFE), although in the popular
press they have been described as such.  IOW, despite what you may have
read, warp bubles are -not- repeat -not- in any sense "a prediction of
gtr"!  Or even "compatible with gtr" as most physicists understand that
term!

To understand this, you need to know that the EFE, the field equation of
gtr, relates two "quantities", like this:

  G_(ab) = 8 Pi T_(ab)

Mathematically speaking, both quantities are "second rank symmetric tensor
fields" defined on a Lorentzian manifold ("spacetime"), which is a "smooth
four-manifold" M equipped with a (Lorentzian) "metric tesnor"  g_(ab), but
they nonetheless have profoundly different natures (which is of course
exactly why the EFE is "interesting"):

1.  The Einstein curvature tensor G_(ab) at the left hand side (LHS) is
completely determined by the Riemann curvature tensor R_(abcd), i.e. by
"the curvature of spacetime" (but not conversely), -completely
independently- of any physical interpretation of any kind, much less a
particular theory of gravitation.  Let me restate this for emphasis: the
tensor field g_(ab) which is part of the purely mathematical definition of
our Lorenzian manifold (M,g), determines the Riemann curvature tensor
R_(abcd), and a kind of "averaging" then produces G_(ab) from R_(abcd).
No physics is anywhere in sight on the LHS of the EFE!

2. The stress-momentum-energy tensor T_(ab) on the right hand side (RHS)
-does- have a direct physical interpretation in gtr.  Basically,

  (a) a "perfect fluid" gives a very particular type of contribution to
T_(ab), which is determined by a theory of perfect fluids (which is
independent of gtr, but can be fixed up to work on curved spacetimes),

  (b) an electromagnetic field gives another very particular (quite
different) type of contribution to T_(ab), according to a prescription
which was determined by Maxwell as part of his theory of EM (again
independent of gtr, but later fixed up to work on curved spacetimes),

and so forth--- at least in principle.  Since gtr is a classical field
theory rather than a quantum field theory, there aren't too many
"realistic" classical field theories to throw into the mix along with
various idealized types of "matter" which might be present in some region
of spacetime!  Be this as it may, the point is that you add up all the
T_(ab) terms coming from all the various types of matter and
nongravitational fields which are present, and put that on the RHS of the
EFE.

Now, a common student error is to think that you can turn this around,
that you can start with any old Lorentzian spacetime, compute the tensor
G_(ab), then divide by 8 Pi and call the result "T_(ab)".  Of course, -if-
you could always interpret this alleged "T_(ab)" as a
stress-momentum-energy tensor, gtr would be entirely -vacuous-!  This is
because this would mean that -any- Lorentzian manifold is a "solution" to
the EFE!  Needless to say, in mathematical physics, equations for which
-anything- is a "solution" are completely useless!

Of course, the point is that requiring that T_(ab) arise "in the usual
way" from matter or nongravitational fields such as an EM field places
-very stringent conditions- upon the our Lorentzian manifold (M,g)
together with any additional vector and tensor fields which might be
defined on M in order to represent the distribution of matter and of any
nongravitational fields.  In fact, solving the EFE is not only not
trivial, but it is quite challenging!

Actually, it's even harder than I've yet indicated.  To see why, suppose
we are seeking a solution where the only nongravitational field is an EM
field, and no "matter" is present.  Such a solution is called an
"electrovacuum solution" or "Einstein-Maxwell" solution.  To find one, you
not only need to find a Lorentzian manifold (M,g) and an EM field F_(ab)
which satisfies the "curved space Maxwell field equation", but when you
compute T_(ab) according to the prescription of Maxwell, and compare with
the G_(ab) computed purely mathematically from the R_(abcd) computed from
the metric tensor g_(ab), then T_(ab) and G_(ab) must satisfy the EFE.
So anything more complicated than a vacuum solution (no matter or
nongravitational fields of any kind) really involves a complicated
(nonlinear!) -simultaneous- solution, not just of the EFE, but of
additional field equations (or equations defining, say, the properties of
a perfect fluid).

(Actually, it's even trickier than -this-, but you get the idea...
suffice it to say, it's a bit of a miracle that tens of thousands of exact
solutions of the EFE are known.)

For more about the EFE, see

  http://math.ucr.edu/home/baez/gr/gr.html

  http://xxx.lanl.gov/abs/gr-qc/0103044

  http://math.ucr.edu/home/baez/PUB/efe

Now, long ago physicists noticed that the T_(ab) terms which arise from
good approximations of various forms of "matter" and from realistic
classical field theories (i.e., EM) satisfy certain "energy conditions".
This is independent of gtr per se.  It turns out that the class of
Lorentzian spacetimes with putative T_(ab) which satisfy these energy
conditions turns out to be a -very- small subset of the class of all
Lorentzian spacetimes.  The point is, no warp bubble spacetime satisfies
any of the energy conditions (when you compute G_(ab), divide by 8 pi, and
try to interpret the result as T_(ab)), so they cannot possibly arise "in
the usual way" from well-understood, realistic theories of states of
matter or nongravitational classical fields such as EM fields.

For more about the T_(ab) terms arising from realistic physical theories
(as incorporated into gtr), see

  http://math.ucr.edu/home/baez/PUB/tensor

For more about the energy conditions, see

  http://math.ucr.edu/home/baez/PUB/dominantenergy

> feasible in terms of physics?

Certainly not in terms of theoretically well-understood physics, much less
experimentally well-tested physics!  The concept of a warp bubble fell
firmly into the realm of "highly speculative theoretical proposals" from
the start, and within a few years had move into the even murkier realm of
"highly dubious and speculative theoretical proposals", as arguments
accumulated that such things cannot exist.

Caveat: it is true that when one attempts with suitable caution to mix up
quantum theory with gtr (this turns out to be very tricky!), then
well-established theory confirmed by replicable experiments (e.g. the
"Casimir force") show that some quantum systems can exhibit a kind of
"negative energy density" and other features which would violate the
classical energy conditions of gtr.  However, this does -not- imply that
such quantum effects can be used to create "warp bubbles"!

> 2) What is exotic matter

A theorist's plaything.

> (matter with a negative mass?),

More like "hypothetical stuff with negative energy density", or even
better, "any hypothetical stuff which, if it existed, would violate at
least one of the classical energy conditions".

"Matter" as that term is commonly understood, of course, has -positive-
energy density.

> and why is it needed for this drive?

In warp bubble spacetimes, G_(ab) is nonzero, so these cannot possibly be
-vacuum- solutions in gtr (the only kind whose definition, G_(ab) = 0, is
in a sense purely mathematical, indepedently of gtr).  Thus, if they were
solutions of the EFE, they would have to be nonvacuum solutions.
However, the Einstein tensors, inside the "walls" of the "bubble", simply
is not compatible with the classical energy conditions.  (This was in fact
noticed by Alcubierre.)  It turns out that this is true not only for the
Alcubierre and Broeck spacetimes, but (at least subject to some
hypotheses) for -any- spacetime containing a superluminal warp bubble.
These results go by the name of "superluminal censorship theorems".

Furthermore, it seems that so-called "quantum inqualities" would
apparently prevent warp bubbles from being realized using something like
the Casimir effect because it would require amazingly stupendous amounts
of energy.  The Van Den Broeck spacetime actually arose in an attempt to
overcome this objection, but Van Den Broeck introduced a new kind of
apparently "unphysical" region in the "walls" of the "bubble" in order to
achieve this trick, and thus this bubble is apparently physically
unrealistic for, if anything, even -more- reasons than Alcubierre's
spacetime, since the alleged "energy reduction" never brought the numbers
into the reasonable realm--- as I recall, it was more like "reducing" the
energy requirement from the mass-energy of several thousand galaxies,
mysteriously turned into -negative- mass-energy and concentrated in a tiny
-tiny- region, to the mass energy of several thousand suns, all this for a
bubble which could move a busload of people from Paris to Beijing at an
"effective speed" of twice the speed of light.

Also, it turns out that (at least subject to some technical conditions)
superluminal warp bubbles, if they exist, would behave much like time
machines.  Most contemporary physicists doubt very much that time machines
are physically possible, although as yet no rigorous proof exists, and at
least one prominent physicist (Igor Novikov)  thinks that they -can-
exist, albeit with mysteriously circumscribed capabilities.  But this is
-extremely- speculative, and in my view, any suggestion that time machines
are realizable should be regarded with great sceptism.

And as if this were not enough, the Alcubierre and Broeck spacetimes and
indeed (technical quibbles deleted) apparently -any- spacetime containing
a superluminal warp bubble would require the existence of "tachyonic
energy transport", something for which there is absolutely no experimental
evidence and which is highly suspect even in theory.  Again, most
physicists would immediately dismiss anything requiring tachyonic energy
transport as physically impossible.  This last (warp bubbles need
tachyonic stuff) is the most elementary objection of all, because it rests
upon a simple "eigenthing analysis" of the Einstein tensors of these
spacetimes.

"Superluminal" warp bubble: warp bubble spacetimes allow for the
possibility of a bubble which forms, accelerates away and eventually in a
sense "goes superluminal", accompanied by various phenonena somewhat
analogous to a sonic boom.  The above arguments focus on showing that even
if something like a warp bubble could be -formed- and even if it could
then be made to -move along-, it could never "go superluminal".  I was
interested in trying to study -subluminal- bubbles, which might not be
subject to some of the above objections.  In fact, however, in the
Alcubierre and Broeck spacetimes even subluminal bubbles which never "go
superluminal" suffer from some terrible properties which tend to suggest
that the basic "Alcubierre trick" (a kind of "bump function blending" of
two or more legitimate solutions of the EFE, which apparently almost
always produces a spacetime with "unphysical" properties) may be fatally
flawed.  As a matter of fact, studying the properties of such "blends"
might make a good jumping off point for a Ph.D. student interested in
working in general relativity, -provided- that doing thesis research in
gtr is a good idea, which for a theoretical physics student might not be
true--- but that's another story.

You can also look here

  http://www.lns.cornell.edu/spr/

for previous posts by myself and others summarizing such theoretical
arguments against the physical existence of warp bubbles, together with
references to the approximately one dozen papers which have appeared on
the subject, almost all devoted to arguing that such things are
impossible.  (No new papers have appeared for quite a while so I think the
naysayers have won their case, at least for the forseeable future.)

> 3) If the Casimir Effect

As I said, the Casimir effect -does- exhibit some properties suggestive of
exotic matter.  However, it turns out that so-called "quantum inqualities"
would apparently prevent warp bubbles from being realized because it would
require so much energy.  As I said, there are other, more or less
independent arguments which suggest that warp bubbles cannot be physically
realized.

> a direct distortion of spacetime around the ship itself?

Look up Alcubierre's original paper and then the papers cited in my long
extinct posts here.  In a sense, as MG pointed out, "space" is compressed
in front of the bubble and expanded behind.  However, I stress again that
these are just Lorentzian manifolds, and this is really verbal shorthand
for the behavior in these spacetimes of another quantity, the "expansion
scalar" h(X), which can be defined purely mathematically for any vector
field X (in this case, defining the world lines of a family of observers)
on (M,g).  There is really -no- known physical interpretation known for
these manifolds, and they certainly are -not- in any sense "solutions" of
the EFE.

> 4) Assuming that the theory works, and that the Casimir Effect could
> generate sufficient quantities of exotic matter (after about a
> thousand years of R&D), what is a good ship design that incorporates
> the Alcubierre/Broeck drive? 5)Are there alternative theories for
> obtaining exotic matter, such as mining it around a black hole?

Exotic matter and black holes are completely different concepts.  They
really have little if anything to do with each other, according to current
theory.  Or perhaps I should say, they aren't really comparable concepts
because black holes are theoretically -very well established-, and the
observational evidence for the existence of astrophysical objects with all
the properties of black holes is now accepted as -overwhelming- (and also
steadily growing in size and sophistication), whereas exotic matter is
speculative even in theory, and at present there is absolutely -no-
experimental or astronomical evidence I know of suggestive of warp bubbles
or time machines or tachyonic stuff.

Chris Hillman

Home page: http://www.math.washington.edu/~hillman/personal.html